Final Project Code - Used Cars

data_sci

Author

Hiba Khatib, Emily Leibfritz, Nicole Birova, and YuYan Zhang

Published

May 23, 2023

Abstract
This file contains the code for the project on , as part of the STAT303-3 course in Spring 2023. 
from sklearn.linear_model import LinearRegression
import pandas as pd
import numpy as np 
import seaborn as sns 
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split, cross_val_predict, cross_val_score, GridSearchCV, ParameterGrid, KFold, \
RandomizedSearchCV
from sklearn.preprocessing import StandardScaler, PolynomialFeatures
from sklearn.metrics import accuracy_score, mean_absolute_error, mean_squared_error, r2_score, \
roc_curve, auc, precision_score, recall_score, confusion_matrix
import statsmodels.formula.api as smf
import statsmodels.api as sm
from patsy import dmatrix
from pyearth import Earth
from sklearn.tree import DecisionTreeClassifier, DecisionTreeRegressor
from sklearn.ensemble import BaggingRegressor,BaggingClassifier,RandomForestRegressor,RandomForestClassifier
from sklearn.neighbors import KNeighborsRegressor
import warnings
warnings.filterwarnings('ignore')
from sklearn.preprocessing import LabelEncoder
import random
from sklearn.ensemble import VotingRegressor, StackingRegressor
data = pd.read_csv('us_used_cars_final.csv')
data.head(2)
vin back_legroom body_type city city_fuel_economy daysonmarket dealer_zip description engine_cylinders engine_displacement ... torque transmission transmission_display trimId trim_name wheel_system wheel_system_display wheelbase width year
0 5LMCJ2C97HUL71203 36.8 in SUV / Crossover Devine 21.0 40 78016 *We have 19 sources of financing and second ch... I4 2000.0 ... 270 lb-ft @ 3,000 RPM A Automatic t66989 Select FWD FWD Front-Wheel Drive 105.9 in 84.1 in 2017
1 1FM5K8D87HGE07214 39.5 in SUV / Crossover Buffalo 16.0 9 55313 We are open for business. Purchase your nex... V6 3500.0 ... 255 lb-ft @ 4,000 RPM A 6-Speed Automatic t67141 XLT AWD AWD All-Wheel Drive 112.8 in 90.2 in 2017

2 rows × 57 columns

1 Data quality check / cleaning / preparation

Put code with comments. The comments should explain the code such that it can be easily understood. You may put text (in a markdown cell) before a large chunk of code to explain the overall purpose of the code, if it is not intuitive. Put the name of the person / persons who contributed to each code chunk / set of code chunks. An example is given below.

1.1 Distribution of response

By Hiba Khatib

#...Plot for distribution of response...#

plt.figure(figsize=(10,6))
sns.distplot(data.price)
plt.title('Price')
plt.legend()
plt.xlabel('Price')
plt.ylabel('Frequency')
plt.ticklabel_format(style='plain', axis='x')
plt.show()

# Mean and standard deviation of response #

print("Mean of price: ", np.mean(data.price))
print("Standard deviation: ", np.std(data.price))
No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Mean of price:  24568.3835
Standard deviation:  16742.99685916766

The data needs a log transformation on price.

plt.figure(figsize=(10,6))
sns.distplot(np.log(data.price))
plt.title('Price')
plt.legend()
plt.xlabel('log(Price)')
plt.ylabel('Frequency')
plt.ticklabel_format(style='plain', axis='x')
plt.show()
No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

The distribution is much more uniform now with the log transformation.

# checking for outlying observations 
sns.boxplot(data.price, orient = "v")
<AxesSubplot:xlabel='price'>

data.price.describe()
count      6000.000000
mean      24568.383500
std       16744.392283
min        1680.000000
25%       16500.000000
50%       21924.000000
75%       29995.000000
max      859000.000000
Name: price, dtype: float64

There appears to be many outliers that we can drop before performing any feature selection. Dropping the outliers should help us avoid possible issues with overfitting to the training data and the complexities that are related to outlying observations.

data_subset = data[data.price <= 30000]
sns.boxplot(data_subset.price)
<AxesSubplot:xlabel='price'>

#...Plot for distribution of response after removing outliers...#

plt.figure(figsize=(10,6))
sns.distplot(data_subset.price)
plt.title('Price')
plt.legend()
plt.xlabel('Price')
plt.ylabel('Frequency')
plt.ticklabel_format(style='plain', axis='x')
plt.show()

# Mean and standard deviation of response #

print("Mean of price: ", np.mean(data_subset.price))
print("Standard deviation: ", np.std(data_subset.price))
No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.

Mean of price:  19205.211827007945
Standard deviation:  5978.306973568847

The mean and standard deviation have significantly decreased after removing all of the outliers and now we don’t need to transform the predictor. The mean and standard deviation have also decreased significantly.

data = data_subset.copy()
print("The final shape of the data after removing outliers is", data.shape)
The final shape of the data after removing outliers is (4532, 57)

1.2 Data Cleaning, Preparation, and EDA

By Hiba Khatib & Emily Leibfritz

After creating all 8 base models, we decided to subset the data into two different ways to improve the analysis of our data. The first subset is the resulting features after our EDA and data preparation. The second way we subsetted the data was based on the features selected by tree based models which is summarized later in the report (done by Hiba) Our methods are described in detail below.

  1. Our EDA Selected Features (written and performed by Hiba & Emily)

We performed data cleaning, preparation and EDA on the data to facilitate feature selection and model training. The first thing we did was fix the data types of some of the columns. All of the boolean columns were encoded as True or False so we mapped those to 1s and 0s. Then, we found that several columns such as horsepower and torque were numerical but contained units. We removed all of the units from these columns and converted them back to numeric. Then, we dropped a set of unique identifiers which were not sigificant to our analysis. We also dropped three columns which represented the interior, and exterior color of the car. These columns had different names for the same color and we did not see them as significant predictors to engineer and encode differently. Then, we dropped a constant predictor which contained the same value for all observations as constant predictors don’t provide any meaningful insights. The final step in the cleaning and preparation of the numerical data, we dropped highly correlated features. These were features which represented the same information in two different ways such as the official name of a trim and the id associated with that name. This resulted in us keeping only 14 of the numerical predictors are subsetting for predictors with a correlation of |-.2|.

For the categorical columns, each was observed individually in order to find the best way to handle them, as just converting them all into dummies resulted in a huge dataset.

Some columns with few unique values that showed a connection to the target variable were just converted into dummies, like body_type and transmission_display.

Some columns, like engine_cylinders and transmission display had some values very common while others only had under 100 observations. These were grouped as others, and the columns were converted into dummies.

Some columns like franchise_make and model_name had a too large amount of unique values to convert into dummies, so the observations were grouped by mean value of target variable, and assigned a number between 1-6.

Some columns did not show any connection to the target variable, while often having a big amount of unique values, like the columns concerning the color of the vehicles, city or major_options. These columns were dropped.

Some columns contained the same information given by other columns already, like the pairs trim&trimID, make_name&franchise_make, engine_type&engine_cylinders where each trim, engine_type and make_name were dropped.

# feature engineering

    # encoding True/False as 1/0s
bool_cols = data.dtypes[data.dtypes == bool]
bool_cols = pd.DataFrame(bool_cols).reset_index()
bool_cols = bool_cols['index']
bool_cols = list(bool_cols)
print(bool_cols)
for col in bool_cols:
    data[col] = data[col].map({False: 0, True: 1})
    

    # numerically encoding categorical variables with at most 5 unique values 
cols_map = []
for c in data.columns:
    col_len = (len(data[c].unique()))
    if col_len <=5:
        cols_map.append(c)
[cols_map.remove(c) for c in bool_cols] # removing the boolean columns from the list

    # performing the encoding
le = LabelEncoder()
encoder_dict = {key: None for key in cols_map}
for c in cols_map:
    encoder_dict[c] = data[c].unique()
    col_name = str(c)+"_cat"
    data[col_name] = le.fit_transform(data[c])
    encoder_dict[col_name] = le.fit_transform(encoder_dict[c])
    
    # removing units from numerical columns 

data['front_legroom'] = data['front_legroom'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['width'] = data['width'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['height'] = data['height'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['length'] = data['length'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['wheelbase'] = data['wheelbase'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['back_legroom'] = data['back_legroom'].map(lambda x: x.lstrip('+-').rstrip('in'))
data['maximum_seating'] = data['maximum_seating'].map(lambda x: x.lstrip('+-').rstrip('seating'))
data['fuel_tank_volume'] = data['fuel_tank_volume'].map(lambda x: x.lstrip('+-').rstrip('gal'))
data['power'] = data['power'].str[:3] # keeping only numerical value
data['torque'] = data['torque'].str[:3]

    # converting data types to numerical
data.power = pd.to_numeric(data.power)
data.back_legroom = pd.to_numeric(data.back_legroom)
data.front_legroom = pd.to_numeric(data.front_legroom)
data.width = pd.to_numeric(data.width)
data.height = pd.to_numeric(data.height)
data.length = pd.to_numeric(data.length)
data.wheelbase = pd.to_numeric(data.wheelbase)
data.maximum_seating = pd.to_numeric(data.maximum_seating)
data.fuel_tank_volume = pd.to_numeric(data.fuel_tank_volume)
data.torque = pd.to_numeric(data.torque)
['fleet', 'frame_damaged', 'franchise_dealer', 'has_accidents', 'isCab', 'is_new', 'salvage', 'theft_title']
data.columns
Index(['vin', 'back_legroom', 'body_type', 'city', 'city_fuel_economy',
       'daysonmarket', 'dealer_zip', 'description', 'engine_cylinders',
       'engine_displacement', 'engine_type', 'exterior_color', 'fleet',
       'frame_damaged', 'franchise_dealer', 'franchise_make', 'front_legroom',
       'fuel_tank_volume', 'fuel_type', 'has_accidents', 'height',
       'highway_fuel_economy', 'horsepower', 'interior_color', 'isCab',
       'is_new', 'latitude', 'length', 'listed_date', 'listing_color',
       'listing_id', 'longitude', 'main_picture_url', 'major_options',
       'make_name', 'maximum_seating', 'mileage', 'model_name', 'owner_count',
       'power', 'price', 'salvage', 'savings_amount', 'seller_rating', 'sp_id',
       'sp_name', 'theft_title', 'torque', 'transmission',
       'transmission_display', 'trimId', 'trim_name', 'wheel_system',
       'wheel_system_display', 'wheelbase', 'width', 'year', 'fuel_type_cat',
       'transmission_cat', 'wheel_system_cat', 'wheel_system_display_cat'],
      dtype='object')
# dropping unique features

data.drop(columns = ['vin', 'listing_id', 'main_picture_url', 'description', 'sp_name', 'make_name', 'city'], inplace = True)

    # color columns
data.drop(columns = ['listing_color', 'interior_color', 'exterior_color'], inplace = True)

print("Data shape after dropping unique features: ", data.shape)

# dropping constant predictor 
print(data.nunique() == 1)
data.drop(columns = ['franchise_dealer'], inplace = True)

# dropping highly correlated columns 

data.drop(columns = ['trim_name', 'engine_type', 'major_options', 'horsepower'], inplace = True)
Data shape after dropping unique features:  (4532, 51)
back_legroom                False
body_type                   False
city_fuel_economy           False
daysonmarket                False
dealer_zip                  False
engine_cylinders            False
engine_displacement         False
engine_type                 False
fleet                       False
frame_damaged               False
franchise_dealer             True
franchise_make              False
front_legroom               False
fuel_tank_volume            False
fuel_type                   False
has_accidents               False
height                      False
highway_fuel_economy        False
horsepower                  False
isCab                       False
is_new                      False
latitude                    False
length                      False
listed_date                 False
longitude                   False
major_options               False
maximum_seating             False
mileage                     False
model_name                  False
owner_count                 False
power                       False
price                       False
salvage                     False
savings_amount              False
seller_rating               False
sp_id                       False
theft_title                 False
torque                      False
transmission                False
transmission_display        False
trimId                      False
trim_name                   False
wheel_system                False
wheel_system_display        False
wheelbase                   False
width                       False
year                        False
fuel_type_cat               False
transmission_cat            False
wheel_system_cat            False
wheel_system_display_cat    False
dtype: bool
print("Final data shape after cleaning and dropping unnecessary features: ", data.shape)
Final data shape after cleaning and dropping unnecessary features:  (4532, 46)

1.4 Feature Selection

final_corrs = pd.DataFrame(data.corrwith(data.price)).reset_index().sort_values(by = 0, ascending = False)

final_corr_cols = list(final_corrs.loc[(final_corrs[0] >= 0.2) | (final_corrs[0] <= -0.2)][1:]['index'])
final_corrs.loc[(final_corrs[0] >= 0.2) | (final_corrs[0] <= -0.2)][1:]
index 0
21 model_name 0.581373
34 year 0.574644
31 trimId 0.570613
23 power 0.406592
30 torque 0.386623
33 width 0.303995
53 transmission_display_8_Speed_Automatic 0.268433
32 wheelbase 0.256636
11 height 0.245438
48 engine_cylinders_V6 0.245117
16 length 0.238048
9 fuel_tank_volume 0.225640
4 engine_displacement 0.220415
26 savings_amount 0.203765
7 franchise_make 0.202163
1 city_fuel_economy -0.203095
37 wheel_system_cat -0.203731
38 wheel_system_display_cat -0.208184
12 highway_fuel_economy -0.230709
44 body_type_Sedan -0.231848
47 engine_cylinders_I4 -0.232696
22 owner_count -0.383574
20 mileage -0.566824 
X = data.copy()

for col in X:
    if col not in final_corr_cols:
        X.drop(col, axis = 1, inplace = True)

X.shape
(4532, 23)

We have 23 predictors which have a reasonable correlation with price which we will be the only predictors we use in our model building.

When we develop our models, we will print the feature importances and note which features are deemed significant by each model.

# checking for any new NAs 
X.isna().sum()
city_fuel_economy                         0
engine_displacement                       0
franchise_make                            0
fuel_tank_volume                          1
height                                    1
highway_fuel_economy                      0
length                                    1
mileage                                   0
model_name                                0
owner_count                               0
power                                     0
savings_amount                            0
torque                                    0
trimId                                    0
wheelbase                                 1
width                                     1
year                                      0
wheel_system_cat                          0
wheel_system_display_cat                  0
body_type_Sedan                           0
engine_cylinders_I4                       0
engine_cylinders_V6                       0
transmission_display_8_Speed_Automatic    0
dtype: int64
y = data.price 
y.isna().sum()
0

The original dataset had ‘–’ for this one observation. When we performed the data cleaning and prep, these observations were not considered missing but became missing after our feature engineering. We can drop this observation since it is only one row with data missing from the original source.

X.isna().sum()
X.iloc[29]
X.drop(29, inplace = True)
y.drop(29, inplace = True)
X.to_csv('X.csv')
y.to_csv('y.csv')
clean_X = X.copy()
clean_y = y.copy()
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state = 4)

1.5 Features Selection using Tree Based Models

Using the tree based methods such as RandomForest and DecisionTrees will allow us to double check if our feature selection has been significant or if it produces better results for base models.

The original shape of our data after removing the outliers is 4532, 58.

After our data cleaning and dropping of the units, we created some NaN values which can be dropped. We can confidently drop these observations because they were initially missing but encoded as ‘–’ which is why we did not initially discover that they were missing. These are 31 observations which we can drop which brings our data shape to 4301, 58

data.shape
(4532, 58)
data_copy = data.copy()
data_copy.dropna(inplace = True)
data_copy.shape
(4501, 58)
X_tree = data_copy.drop(columns = ['price'])
y_tree = data_copy.price

1.5.1 Base Random Forest Feature Selection

rf_features = RandomForestRegressor(random_state = 4).fit(X_tree, y_tree)
rf_feats = pd.DataFrame(zip(rf_features.feature_names_in_, rf_features.feature_importances_))
rf_feats.sort_values(by = 1, ascending = False)
0 1
30 trimId 3.681064e-01
21 model_name 2.059636e-01
23 power 1.224561e-01
20 mileage 8.733049e-02
29 torque 2.501834e-02
0 back_legroom 1.947877e-02
25 savings_amount 1.366988e-02
11 height 1.346977e-02
47 engine_cylinders_V6 9.795723e-03
33 year 9.452366e-03
15 latitude 9.071714e-03
26 seller_rating 8.884898e-03
27 sp_id 8.311864e-03
16 length 8.101590e-03
12 highway_fuel_economy 8.033217e-03
32 width 7.344786e-03
9 fuel_tank_volume 7.089915e-03
2 daysonmarket 6.167395e-03
36 wheel_system_cat 6.084273e-03
18 longitude 5.956792e-03
37 wheel_system_display_cat 5.825781e-03
31 wheelbase 5.460589e-03
3 dealer_zip 5.311008e-03
8 front_legroom 5.194709e-03
17 listed_date 4.584625e-03
4 engine_displacement 4.355540e-03
1 city_fuel_economy 3.154374e-03
7 franchise_make 2.225961e-03
42 body_type_SUV_Crossover 1.811286e-03
22 owner_count 1.508617e-03
19 maximum_seating 1.295084e-03
51 transmission_display_6_Speed_Automatic 9.956286e-04
10 has_accidents 9.003557e-04
54 transmission_display_Automatic 8.624836e-04
13 isCab 8.238435e-04
5 fleet 7.241308e-04
52 transmission_display_8_Speed_Automatic 6.188337e-04
43 body_type_Sedan 5.821297e-04
46 engine_cylinders_I4 5.186393e-04
34 fuel_type_cat 5.060303e-04
35 transmission_cat 5.033826e-04
55 transmission_display_Continuously_Variable_Tra... 4.730021e-04
56 transmission_display_other 3.798145e-04
45 body_type_Wagon 2.671794e-04
53 transmission_display_9_Speed_Automatic 2.053301e-04
48 engine_cylinders_V6_Flex_Fuel_Vehicle 1.959885e-04
39 body_type_Hatchback 1.790901e-04
14 is_new 1.749637e-04
41 body_type_Pickup_Truck 1.455960e-04
40 body_type_Minivan 1.367203e-04
50 engine_cylinders_other 1.187689e-04
49 engine_cylinders_V8 8.934867e-05
38 body_type_Coupe 6.330553e-05
28 theft_title 1.080973e-05
6 frame_damaged 6.621246e-06
44 body_type_Van 2.167886e-06
24 salvage 4.001456e-07 
rf_feats[(rf_feats[1] >= 0.2) | (rf_feats[1] <= -.2)]
0 1
21 model_name 0.205964
30 trimId 0.368106 
base_dt = DecisionTreeRegressor(random_state = 4).fit(X_tree, y_tree)
dt_feats = pd.DataFrame(zip(base_dt.feature_names_in_, base_dt.feature_importances_))
dt_feats.sort_values(by = 1, ascending = False)
dt_feats[(dt_feats[1] >= 0.2) | (dt_feats[1] <= -.2)]
0 1
21 model_name 0.211757
30 trimId 0.379266 
bagged_dt = BaggingRegressor(DecisionTreeRegressor(), random_state = 4, n_estimators = 100).fit(X_tree, y_tree)
f_imp = pd.DataFrame(zip(bagged_dt.feature_names_in_,np.mean([tree.feature_importances_ for tree in bagged_dt.estimators_], axis = 0)))
f_imp.sort_values(by = 1, ascending = False)
f_imp[(f_imp[1] >= 0.2) | (f_imp[1] <= -.2)]
0 1
21 model_name 0.205927
30 trimId 0.368145

Based on the base Random Forest and DecisionTree, the two most significant predictors are model_name and trimId. We can test the performance of a model developed using our feature selection and a model developed using the selected features by the RandomForest to find the best predictors.

X_tree_subset = X_tree[['model_name', 'trimId']]

X_train_sub, X_test_sub, y_train_sub, y_test_sub = train_test_split(X_tree_subset, y_tree, test_size = 0.3, random_state = 4)

Base MARS model on the tree selected features:

base_mars1 = Earth().fit(X_train_sub, y_train_sub)
print(base_mars1.summary())
print("Base MARS train RMSE", np.sqrt(base_mars1.mse_))
print("\nBase MARS train R^2", base_mars1.rsq_)
print("\nBASE MARS Test RMSE", mean_squared_error(y_test_sub, base_mars1.predict(X_test_sub), squared = False))
print("\nBASE MARS Test R^2", r2_score(y_test_sub, base_mars1.predict(X_test_sub)))
Earth Model
--------------------------------------
Basis Function   Pruned  Coefficient  
--------------------------------------
(Intercept)      No      6604.08      
h(trimId-67312)  No      -0.228409    
h(67312-trimId)  Yes     None         
model_name       No      3970.31      
h(trimId-32660)  No      0.196858     
h(32660-trimId)  Yes     None         
h(trimId-83763)  Yes     None         
h(83763-trimId)  Yes     None         
h(trimId-89881)  No      -2.24943     
h(89881-trimId)  Yes     None         
h(trimId-78253)  No      -6.47566     
h(78253-trimId)  Yes     None         
h(trimId-76583)  No      2.17212      
h(76583-trimId)  Yes     None         
h(trimId-78985)  No      5.02076      
h(78985-trimId)  No      -1.16045     
h(trimId-82290)  Yes     None         
h(82290-trimId)  No      1.05506      
trimId           Yes     None         
--------------------------------------
MSE: 12520924.7798, GCV: 12709856.5249, RSQ: 0.6517, GRSQ: 0.6466
Base MARS train RMSE 3538.4918792925073

Base MARS train R^2 0.651676960724981

BASE MARS Test RMSE 3596.9618257602892

BASE MARS Test R^2 0.6314408357383765

Base MARS model on our selected features:

base_mars2 = Earth().fit(X_train, y_train)
print(base_mars2.summary())
print("Base MARS train RMSE", np.sqrt(base_mars2.mse_))
print("\nBase MARS train R^2", base_mars2.rsq_)
print("\nBASE MARS Test RMSE", mean_squared_error(y_test, base_mars2.predict(X_test), squared = False))
print("\nBASE MARS Test R^2", r2_score(y_test, base_mars2.predict(X_test)))
Earth Model
-----------------------------------------------
Basis Function            Pruned  Coefficient  
-----------------------------------------------
(Intercept)               No      -696322      
h(trimId-67312)           Yes     None         
h(67312-trimId)           No      -0.126974    
model_name                No      2187.29      
power                     No      14.7161      
h(mileage-116656)         No      -0.032541    
h(116656-mileage)         No      0.0639984    
wheel_system_display_cat  No      -355.43      
engine_cylinders_V6       No      1588.48      
torque                    No      11.9225      
h(trimId-37340)           Yes     None         
h(37340-trimId)           No      0.187547     
wheel_system_cat          No      -399.524     
h(savings_amount-4356)    No      2.82244      
h(4356-savings_amount)    No      -0.259224    
year                      No      351.544      
body_type_Sedan           No      -761.063     
h(city_fuel_economy-32)   Yes     None         
h(32-city_fuel_economy)   No      -334.968     
length                    No      67.983       
franchise_make            No      223.224      
highway_fuel_economy      No      -175.368     
wheelbase                 No      -67.8022     
-----------------------------------------------
MSE: 7209072.4230, GCV: 7434760.4238, RSQ: 0.7982, GRSQ: 0.7921
Base MARS train RMSE 2684.9715869996703

Base MARS train R^2 0.7982451686817187

BASE MARS Test RMSE 2874.664784272254

BASE MARS Test R^2 0.7688754274179378

The model’s performance is much better using the features selected by our analysis than the tree based models. The R^2 is much greater on our subset’s test data even though there does appear to be some slight overfitting. Overfitting does not appear to be an issue with the tree based models selected features but the model’s performance is much worse.

The main takeaway from this analysis is that our selected features resulted in overfitting but generally well performing model. This was true for our initial model development in which most of our models overfit even with extensive tuning and generally had much higher RMSE’s at the base model and tuned model.

Another method for finding the best features is by running a base MARS model on our original data to find the best overall features.

X_tree.shape
(4501, 57)
mars_feats = Earth().fit(X_tree, y_tree)
print(mars_feats.summary())
Earth Model
-------------------------------------------------------------
Basis Function                          Pruned  Coefficient  
-------------------------------------------------------------
(Intercept)                             No      -899382      
h(trimId-67983)                         Yes     None         
h(67983-trimId)                         No      -0.068928    
model_name                              No      2020.83      
power                                   No      16.0908      
h(mileage-117301)                       No      -0.0349422   
h(117301-mileage)                       No      0.0597747    
wheel_system_cat                        No      -358.023     
engine_cylinders_V6_Flex_Fuel_Vehicle   No      -3210.91     
fleet                                   No      -1076.01     
h(savings_amount-1580)                  No      0.804517     
h(1580-savings_amount)                  Yes     None         
h(latitude-30.4119)                     No      -41.4752     
h(30.4119-latitude)                     No      -369.663     
engine_cylinders_I4                     No      -865.66      
wheel_system_display_cat                No      -246.752     
h(trimId-35944)                         No      0.0450436    
h(35944-trimId)                         No      0.114317     
year                                    No      455.331      
transmission_display_6_Speed_Automatic  No      -724.604     
h(daysonmarket-215)                     Yes     None         
h(215-daysonmarket)                     No      59.4532      
highway_fuel_economy                    No      -275.857     
h(city_fuel_economy-32)                 No      -783.609     
h(32-city_fuel_economy)                 No      -264.844     
h(length-231.9)                         No      -910.335     
h(231.9-length)                         No      -55.2657     
seller_rating                           No      437.54       
franchise_make                          No      192.154      
h(daysonmarket-256)                     Yes     None         
h(256-daysonmarket)                     No      -50.5369     
torque                                  No      9.83559      
h(wheelbase-145)                        No      709.676      
h(145-wheelbase)                        No      48.9592      
-------------------------------------------------------------
MSE: 6766485.4663, GCV: 6993008.2321, RSQ: 0.8104, GRSQ: 0.8042
print("Base MARS train RMSE", np.sqrt(mars_feats.mse_))
print("\nBase MARS train R^2", mars_feats.rsq_)
Base MARS train RMSE 2601.2469060613644

Base MARS train R^2 0.8104297884139036
print("Checking performance on split data: \n")
X_train_mars, X_test_mars, y_train_mars, y_test_mars = train_test_split(X_tree, y_tree, random_state = 4, test_size = 0.3)
random.seed(4)
mars_feats2 = Earth(feature_importance_type = 'rss').fit(X_train_mars, y_train_mars)
print("Base MARS Train RMSE:", np.sqrt(mars_feats2.mse_))
print("\nBase MARS Train R^2", mars_feats2.rsq_)
print("\nBase MARS Test RMSE:", mean_squared_error(y_test_mars, mars_feats2.predict(X_test_mars), squared = False))
print("\nBase MARS Test R^2", r2_score(y_test_mars, mars_feats2.predict(X_test_mars)))
Checking performance on split data: 

Base MARS Train RMSE: 2629.4932798721816

Base MARS Train R^2 0.8076510034139952

Base MARS Test RMSE: 2619.6932550051956

Base MARS Test R^2 0.8045046459515068
print(mars_feats2.summary_feature_importances(sort_by = 'rss'))
                                                            rss
model_name                                                  0.43   
mileage                                                     0.36   
power                                                       0.09   
year                                                        0.06   
wheel_system_cat                                            0.02   
engine_cylinders_V6_Flex_Fuel_Vehicle                       0.01   
fleet                                                       0.01   
savings_amount                                              0.01   
engine_cylinders_I4                                         0.00   
city_fuel_economy                                           0.00   
trimId                                                      0.00   
listed_date                                                 0.00   
latitude                                                    0.00   
transmission_display_6_Speed_Automatic                      0.00   
seller_rating                                               0.00   
highway_fuel_economy                                        0.00   
franchise_make                                              0.00   
length                                                      0.00   
wheelbase                                                   0.00   
wheel_system_display_cat                                    0.00   
longitude                                                   0.00   
maximum_seating                                             0.00   
transmission_display_other                                  0.00   
is_new                                                      0.00   
owner_count                                                 0.00   
height                                                      0.00   
has_accidents                                               0.00   
fuel_tank_volume                                            0.00   
front_legroom                                               0.00   
frame_damaged                                               0.00   
engine_displacement                                         0.00   
dealer_zip                                                  0.00   
daysonmarket                                                0.00   
isCab                                                       0.00   
theft_title                                                 0.00   
salvage                                                     0.00   
sp_id                                                       0.00   
transmission_display_Automatic                              0.00   
transmission_display_9_Speed_Automatic                      0.00   
transmission_display_8_Speed_Automatic                      0.00   
engine_cylinders_other                                      0.00   
engine_cylinders_V8                                         0.00   
engine_cylinders_V6                                         0.00   
body_type_Wagon                                             0.00   
body_type_Van                                               0.00   
body_type_Sedan                                             0.00   
body_type_SUV_Crossover                                     0.00   
body_type_Pickup_Truck                                      0.00   
body_type_Minivan                                           0.00   
body_type_Hatchback                                         0.00   
body_type_Coupe                                             0.00   
transmission_cat                                            0.00   
fuel_type_cat                                               0.00   
width                                                       0.00   
torque                                                      0.00   
transmission_display_Continuously_Variable_Transmission     0.00   
back_legroom                                                0.00

The selected features by MARS result in the best performance on the training data and the testing data. There is no overfitting or underfitting and the R^2 has improved, albeit marginally.

Thus, we will perform the rest of our analysis with the top features selected by the best MARS model. These features are:

model_name : 0.43
mileage 0.36
power 0.09
year 0.06
wheel_system_cat 0.02
engine_cylinders_V6_Flex_Fuel_Vehicle 0.01
fleet 0.01
savings_amount 0.01

Final data subsetting and split:

# dropping the 31 missing values from the main/original data

data.dropna(inplace = True)
y_mars = data.price
data.drop(columns = ['price'], inplace = True)
print("Final overall training data shape", data.shape)
print("Final response variable shape", y_mars.shape)

# subsetting for the final selected features

X_mars= data[['model_name', 'mileage', 'power', 'year', 'wheel_system_cat', 'engine_cylinders_V6_Flex_Fuel_Vehicle', 'fleet', 'savings_amount']]

# splitting the final data! 

X_train_mars, X_test_mars, y_train_mars, y_test_mars = train_test_split(X_mars, y_mars, test_size = 0.3, random_state = 4)
Final overall training data shape (4501, 57)
Final response variable shape (4501,)

We will develop our models using both the features found during our EDA and the models using the features found based on the MARS model analysis.

2 Developing the model: Hyperparameter tuning

Throughout our model development, we ran our analysis on both our selected features and the random forest’s selected features to observe the difference in performance. Generally, we anticipate that the models developed using the features selected by random forest will perform better.

X_train, y_train, X_test, and y_test represent the data which contains the feature selection resulting from our EDA and analysis while X_train_mars, y_train_mars, X_test_mars, and y_test_mars represent the data which contains the features selected by the Random Forest model.

2.1 MARS & Bagging MARS

By Hiba Khatib

2.1.1 Tuning MARS models

Since we already performed base MARS model analysis to inform our feature selection, we can continue with tuning MARS to achieve better results. The first set of tuning will be on the MARS selected features then we will also tuning on our selected features.

random.seed(4)
opt_mars = pd.DataFrame(columns = {'degree', 'R2', 'RMSE'})
iter_num = 0
cv = KFold(n_splits=5, random_state=4, shuffle=True)
for i in range(1, 8):
    
    MARS_model = Earth(max_terms=500, max_degree=i)
    r2 = cross_val_score(MARS_model, X_train_mars, y_train_mars, cv=5, scoring='r2')
    
    mean_r2 = r2.mean()
    
    rmse = cross_val_score(MARS_model, X_train_mars, y_train_mars, cv= 5, scoring='neg_root_mean_squared_error')
    mean_rmse = abs(rmse).mean()
    
    
    opt_mars.loc[iter_num, 'degree'] = i
    opt_mars.loc[iter_num, 'R2'] = mean_r2
    opt_mars.loc[iter_num, 'RMSE'] = mean_rmse
    iter_num += 1
opt_mars.sort_values(by = 'R2', ascending = False).head(1)
R2 RMSE degree
4 0.790333 2742.648575 5 
opt_mars.sort_values(by = 'RMSE', ascending = True).head(1)
R2 RMSE degree
4 0.790333 2742.648575 5

The optimal degree is 5 for both R2 and RMSE.

fit_mars1 = Earth(max_terms=1000, max_degree= 5, feature_importance_type = "rss").fit(X_train_mars, y_train_mars)
print(fit_mars1.summary())
Earth Model
----------------------------------------------------------------------------------------------
Basis Function                                                          Pruned  Coefficient   
----------------------------------------------------------------------------------------------
(Intercept)                                                             No      -0.867045     
year                                                                    No      -872.929      
model_name*year                                                         No      1.1003        
power*year                                                              No      0.0247296     
h(mileage-118799)*year                                                  No      3.49146e-05   
h(118799-mileage)*year                                                  Yes     None          
wheel_system_cat*year                                                   No      -0.432392     
engine_cylinders_V6_Flex_Fuel_Vehicle*model_name*year                   Yes     None          
h(mileage-86269)*model_name*year                                        No      -8.66159e-06  
h(86269-mileage)*model_name*year                                        Yes     None          
fleet*power*year                                                        No      -0.00229866   
year*year                                                               No      0.435352      
h(savings_amount-1293)                                                  No      0.737447      
h(1293-savings_amount)                                                  Yes     None          
power*h(86269-mileage)*model_name*year                                  No      4.76397e-06   
mileage*power*year                                                      No      -1.1364e-07   
wheel_system_cat*engine_cylinders_V6_Flex_Fuel_Vehicle*model_name*year  No      -0.503445     
power*power*h(86269-mileage)*model_name*year                            No      -4.14987e-10  
year*power*h(86269-mileage)*model_name*year                             No      -2.30134e-09  
----------------------------------------------------------------------------------------------
MSE: 7320763.8309, GCV: 7491008.4641, RSQ: 0.7963, GRSQ: 0.7917
print("Performance on MARS feature selection:\n")
print("Tuned MARS Train RMSE:", np.sqrt(fit_mars1.mse_))
print("Tuned MARS Train R^2 value: ", fit_mars1.rsq_)
print("Tuned MARS Test RMSE:",mean_squared_error(y_test_mars, fit_mars1.predict(X_test_mars), squared = False) )
print("Tuned MARS Test R^2:", r2_score(y_test_mars, fit_mars1.predict(X_test_mars)))
Performance on MARS feature selection:

Tuned MARS Train RMSE: 2705.691008032262
Tuned MARS Train R^2 value:  0.7963416638745415
Tuned MARS Test RMSE: 2679.9773361980965
Tuned MARS Test R^2: 0.7954036885971033
print("Feature importance by RSS on MARS feature selection\n")
print(fit_mars1.summary_feature_importances(sort_by = 'rss'))
Feature importance by RSS on MARS feature selection

                                          rss
year                                      0.59   
model_name                                0.24   
power                                     0.09   
mileage                                   0.05   
wheel_system_cat                          0.02   
engine_cylinders_V6_Flex_Fuel_Vehicle     0.01   
fleet                                     0.00   
savings_amount                            0.00

Performance on our feature selection:

random.seed(4)
opt_mars2 = pd.DataFrame(columns = {'degree', 'R2', 'RMSE'})
iter_num = 0
cv = KFold(n_splits=5, random_state=4, shuffle=True)
for i in range(1, 8):
    
    MARS_model = Earth(max_terms=500, max_degree=i)
    r2 = cross_val_score(MARS_model, X_train, y_train, cv=5, scoring='r2')
    
    mean_r2 = r2.mean()
    
    rmse = cross_val_score(MARS_model, X_train, y_train, cv= 5, scoring='neg_root_mean_squared_error')
    mean_rmse = abs(rmse).mean()
    
    
    opt_mars2.loc[iter_num, 'degree'] = i
    opt_mars2.loc[iter_num, 'R2'] = mean_r2
    opt_mars2.loc[iter_num, 'RMSE'] = mean_rmse
    iter_num += 1
print(opt_mars2.sort_values(by = 'R2', ascending = False).head(1))
print(opt_mars2.sort_values(by = 'RMSE', ascending = True).head(1))
        R2         RMSE degree
5  0.81581  2560.543379      6
        R2         RMSE degree
5  0.81581  2560.543379      6

The best degree for both R^2 and RMSE is 6

fit_mars2 = Earth(max_terms=1000, max_degree= 6, feature_importance_type = "rss").fit(X_train, y_train)
print(fit_mars2.summary())
Earth Model
----------------------------------------------------------------------------
Basis Function                                        Pruned  Coefficient   
----------------------------------------------------------------------------
(Intercept)                                           No      7265.17       
h(trimId-67312)                                       No      -0.385114     
h(67312-trimId)                                       No      -0.0784129    
model_name                                            No      -460005       
power                                                 No      50.5903       
h(mileage-116656)                                     No      -0.0256394    
h(116656-mileage)                                     No      0.0538941     
wheel_system_display_cat                              Yes     None          
h(trimId-60496)*h(67312-trimId)                       No      0.00181749    
h(60496-trimId)*h(67312-trimId)                       Yes     None          
h(trimId-67312)*power                                 No      0.00396033    
h(67312-trimId)*power                                 No      -0.000471043  
power*h(trimId-67312)*power                           No      -1.83312e-05  
city_fuel_economy*power*h(trimId-67312)*power         No      4.80116e-07   
highway_fuel_economy*power                            No      -0.7356       
engine_cylinders_V6*h(116656-mileage)                 Yes     None          
torque                                                No      13.6581       
h(trimId-16075)*wheel_system_display_cat              No      -0.0120157    
h(16075-trimId)*wheel_system_display_cat              Yes     None          
h(city_fuel_economy-32)*highway_fuel_economy*power    Yes     None          
h(32-city_fuel_economy)*highway_fuel_economy*power    No      -0.0412428    
h(savings_amount-1465)*wheel_system_display_cat       No      0.262227      
h(1465-savings_amount)*wheel_system_display_cat       Yes     None          
year*model_name                                       No      229.382       
model_name*engine_cylinders_V6*h(116656-mileage)      No      -0.0168493    
engine_displacement*h(trimId-60496)*h(67312-trimId)   No      3.33867e-07   
model_name*h(16075-trimId)*wheel_system_display_cat   No      0.161783      
height*h(trimId-60496)*h(67312-trimId)                No      -4.5658e-05   
height*engine_cylinders_V6*h(116656-mileage)          No      0.000785742   
franchise_make*h(116656-mileage)                      No      0.00241208    
h(engine_displacement-6000)*wheel_system_display_cat  No      6.56446       
h(6000-engine_displacement)*wheel_system_display_cat  Yes     None          
----------------------------------------------------------------------------
MSE: 6085273.5118, GCV: 6326329.5161, RSQ: 0.8297, GRSQ: 0.8231
print("Performance on MARS feature selection:\n")
print("Tuned MARS Train RMSE:", np.sqrt(fit_mars2.mse_))
print("Tuned MARS Train R^2 value: ", fit_mars2.rsq_)
print("Tuned MARS Test RMSE:",mean_squared_error(y_test, fit_mars2.predict(X_test), squared = False) )
print("Tuned MARS Test R^2:", r2_score(y_test, fit_mars2.predict(X_test)))
Performance on MARS feature selection:

Tuned MARS Train RMSE: 2466.834715144428
Tuned MARS Train R^2 value:  0.829696074769481
Tuned MARS Test RMSE: 2743.978978341708
Tuned MARS Test R^2: 0.7894121729348178
print("Feature importance by RSS on MARS feature selection\n")
print(fit_mars2.summary_feature_importances(sort_by = 'rss'))
Feature importance by RSS on MARS feature selection

                                           rss
model_name                                 0.27   
year                                       0.26   
mileage                                    0.24   
power                                      0.11   
trimId                                     0.07   
wheel_system_display_cat                   0.02   
engine_cylinders_V6                        0.01   
highway_fuel_economy                       0.01   
height                                     0.01   
city_fuel_economy                          0.00   
engine_displacement                        0.00   
savings_amount                             0.00   
torque                                     0.00   
franchise_make                             0.00   
owner_count                                0.00   
wheelbase                                  0.00   
width                                      0.00   
length                                     0.00   
wheel_system_cat                           0.00   
body_type_Sedan                            0.00   
fuel_tank_volume                           0.00   
engine_cylinders_I4                        0.00   
transmission_display_8_Speed_Automatic     0.00

Results and Takeaways:

The MARS selected features model seems to outperform the model developed on our subset. There is less overfitting despite the slightly worse R^2. This makes sense because the original MARS feature selection ensured that we kept the features which best represented the data and thus it is less likely to experience over or underfitting as in our selected features. The pruning ensures that the model is performing as optimally as possible with the base parameters.

It is also important to note that the performance on the model with the tuned degree is not significantly better than the performance on the base MARS model with the MARS selected features.

2.1.2 Bagging MARS

First we will bag the best features selected by MARS. We will do a base MARS fitted with the MARS feature data and then a the best MARS feature.

# bagging the base model for reference
bagging_base = BaggingRegressor(base_estimator = Earth(feature_importance_type = 'rss'), 
                                random_state = 4).fit(X_train_mars, y_train_mars)
best_rmse = 10000
best_r2 = 0
for estimator in bagging_base.estimators_:
    if np.sqrt(estimator.mse_) < best_rmse:
        best_rmse = np.sqrt(estimator.mse_)
    if estimator.rsq_ > best_r2:
        best_r2 = estimator.rsq_
print("Bagged base MARS model Train RMSE:", best_rmse)
print("R^2 on Train data", best_r2)
print("Bagged base MARS model Test RMSE:", mean_squared_error(y_test_mars, bagging_base.predict(X_test_mars), squared = False))
print("R^2 on Test data:", r2_score(y_test_mars, bagging_base.predict(X_test_mars)))
print("Feature importances show the same feature importances as the unbagged MARS models.")
# for estimator in bagging_base.estimators_:
#     print(estimator.summary_feature_importances(sort_by = "rss"))
Bagged base MARS model Train RMSE: 2761.6200070905847
R^2 on Train data 0.9588173903334842
Bagged base MARS model Test RMSE: 2788.600146839992
R^2 on Test data: 0.7784824962191642
Feature importances show the same feature importances as the unbagged MARS models.

Now we will bag the best MARS model found on the MARS selected features and also tune the bagging grid for this model.

# bagging the best estimator found on the MARS selected features

bagging_mars = BaggingRegressor(base_estimator = Earth(max_terms=1000, max_degree= 5, feature_importance_type = "rss")).fit(X_train_mars, y_train_mars)
best_rmse = 10000
best_r2 = 0
for estimator in bagging_mars.estimators_:
    if np.sqrt(estimator.mse_) < best_rmse:
        best_rmse = np.sqrt(estimator.mse_)
    if estimator.rsq_ > best_r2:
        best_r2 = estimator.rsq_
print("Bagged base MARS model Train RMSE:", best_rmse)
print("R^2 on Train data", best_r2)
print("Bagged base MARS model Test RMSE:", mean_squared_error(y_test_mars, bagging_mars.predict(X_test_mars), squared = False))
print("R^2 on Test data:", r2_score(y_test_mars, bagging_mars.predict(X_test_mars)))
print("Feature importances are consistent.")
Bagged base MARS model Train RMSE: 2612.6987983602867
R^2 on Train data 0.963612437865765
Bagged base MARS model Test RMSE: 2737.9654132203773
R^2 on Test data: 0.7864539845171037
Feature importances are consistent.
# coarse grid search: optimizing bagging model 

bagging_opt = {'bootstrap': [True, False], 
              'bootstrap_features': [True, False],
               'max_samples': [0.5, 0.75,1.0],
              'max_features': [0.5, 0.75,1.0]}

opt_grid = RandomizedSearchCV(estimator =  BaggingRegressor(base_estimator = Earth(max_degree= 5, feature_importance_type = "rss"), random_state =4), \
                        param_distributions = bagging_opt, verbose = 1, n_jobs = -1, scoring = 'neg_root_mean_squared_error')
opt_grid.fit(X_train_mars, y_train_mars)
Fitting 5 folds for each of 10 candidates, totalling 50 fits
RandomizedSearchCV(estimator=BaggingRegressor(base_estimator=Earth(feature_importance_type='rss',
                                                                   max_degree=5),
                                              random_state=4),
                   n_jobs=-1,
                   param_distributions={'bootstrap': [True, False],
                                        'bootstrap_features': [True, False],
                                        'max_features': [0.5, 0.75, 1.0],
                                        'max_samples': [0.5, 0.75, 1.0]},
                   scoring='neg_root_mean_squared_error', verbose=1)
print("The best BaggingRegressor coarse grid parameters:", opt_grid.best_params_)

best_rmse = 10000
best_r2 = 0
for estimator in opt_grid.best_estimator_.estimators_:
    if np.sqrt(estimator.mse_) < best_rmse:
        best_rmse = np.sqrt(estimator.mse_)
    if estimator.rsq_ > best_r2:
        best_r2 = estimator.rsq_

print("Tuned BaggingRegressor Train RMSE:", best_rmse)
print("Tuned BaggingRegressor R2:", best_r2)
print("Tuned BaggingRegressor Test RMSE: ", mean_squared_error(y_test_mars, opt_grid.predict(X_test_mars), squared = False))
print("Tuned BaggingRegressor R2", r2_score(y_test_mars, opt_grid.predict(X_test_mars)))
The best BaggingRegressor coarse grid parameters: {'max_samples': 0.75, 'max_features': 0.5, 'bootstrap_features': False, 'bootstrap': False}
Tuned BaggingRegressor Train RMSE: 3040.6124116396736
Tuned BaggingRegressor R2: 0.9284106503760392
Tuned BaggingRegressor Test RMSE:  2914.9556727676986
Tuned BaggingRegressor R2 0.7579531323185931
# fine grid search

bagging_opt2 = {'max_samples': np.arange(0.1, 0.75, .2),
              'max_features': np.arange(0.5, 1, .2)}

opt_grid2 = RandomizedSearchCV(estimator =  BaggingRegressor(base_estimator = Earth(max_degree= 5, feature_importance_type = "rss"), \
                                                             random_state =4, bootstrap_features = False,\
                                                            bootstrap = False),  param_distributions = bagging_opt2, verbose = 1,\
                               n_jobs = -1, scoring = 'neg_root_mean_squared_error', random_state = 4, n_iter = 3)
opt_grid2.fit(X_train_mars, y_train_mars)
print("The best BaggingRegressor fine parameters:", opt_grid2.best_params_)

best_rmse = 10000
best_r2 = 0
for estimator in opt_grid2.best_estimator_.estimators_:
    if np.sqrt(estimator.mse_) < best_rmse:
        best_rmse = np.sqrt(estimator.mse_)
    if estimator.rsq_ > best_r2:
        best_r2 = estimator.rsq_

print("Tuned BaggingRegressor Train RMSE:", best_rmse)
print("Tuned BaggingRegressor R2:", best_r2)
print("Tuned BaggingRegressor Test RMSE: ", mean_squared_error(y_test_mars, opt_grid2.predict(X_test_mars), squared = False))
print("Tuned BaggingRegressor R2", r2_score(y_test_mars, opt_grid2.predict(X_test_mars)))
The best BaggingRegressor fine parameters: {'max_samples': 0.6, 'max_features': 0.9}
Tuned BaggingRegressor Train RMSE: 2620.6858359626726
Tuned BaggingRegressor R2: 0.9624349584512664
Tuned BaggingRegressor Test RMSE:  3145.216344284546
Tuned BaggingRegressor R2 0.718202844680064

Finally we will bag the best MARS model found on our selected features and also tune the bagging grid for this model.

# bagging the base model for reference
bagging_base = BaggingRegressor(base_estimator = Earth(feature_importance_type = 'rss'), 
                                random_state = 4).fit(X_train, y_train)
best_rmse = 10000
best_r2 = 0
for estimator in bagging_base.estimators_:
    if np.sqrt(estimator.mse_) < best_rmse:
        best_rmse = np.sqrt(estimator.mse_)
    if estimator.rsq_ > best_r2:
        best_r2 = estimator.rsq_
print("Bagged base MARS model Train RMSE:", best_rmse)
print("R^2 on Train data", best_r2)
print("Bagged base MARS model Test RMSE:", mean_squared_error(y_test, bagging_base.predict(X_test), squared = False))
print("R^2 on Test data:", r2_score(y_test, bagging_base.predict(X_test)))
print("Feature importances show the same feature importances as the unbagged MARS models.")
# for estimator in bagging_base.estimators_:
#     print(estimator.summary_feature_importances(sort_by = "rss"))
Bagged base MARS model Train RMSE: 2597.9794766802843
R^2 on Train data 0.9632264134685602
Bagged base MARS model Test RMSE: 2869.7450024161185
R^2 on Test data: 0.7696658565959801
Feature importances show the same feature importances as the unbagged MARS models.
# bagging the best estimator found on our selected features

bagging_mars = BaggingRegressor(base_estimator = Earth(max_degree= 6, feature_importance_type = "rss")).fit(X_train, y_train)
best_rmse = 10000
best_r2 = 0
for estimator in bagging_mars.estimators_:
    if np.sqrt(estimator.mse_) < best_rmse:
        best_rmse = np.sqrt(estimator.mse_)
    if estimator.rsq_ > best_r2:
        best_r2 = estimator.rsq_
print("Bagged base MARS model Train RMSE:", best_rmse)
print("R^2 on Train data", best_r2)
print("Bagged base MARS model Test RMSE:", mean_squared_error(y_test, bagging_mars.predict(X_test), squared = False))
print("R^2 on Test data:", r2_score(y_test, bagging_mars.predict(X_test)))
print("Feature importances are consistent.")
Bagged base MARS model Train RMSE: 2280.922799036446
R^2 on Train data 0.9720576756220233
Bagged base MARS model Test RMSE: 2940.2215805350525
R^2 on Test data: 0.7582136243310234
Feature importances are consistent.
# coarse grid search: optimizing bagging model 

bagging_opt = {'max_samples': [0.5, 0.75,1.0],
              'max_features': [0.5, 0.75,1.0]}

opt_grid = RandomizedSearchCV(estimator =  BaggingRegressor(base_estimator = Earth(max_degree= 6, feature_importance_type = "rss"), random_state =4), \
                        param_distributions = bagging_opt, verbose = 1, n_jobs = -1, scoring = 'neg_root_mean_squared_error')
opt_grid.fit(X_train, y_train)
Fitting 5 folds for each of 9 candidates, totalling 45 fits
RandomizedSearchCV(estimator=BaggingRegressor(base_estimator=Earth(feature_importance_type='rss',
                                                                   max_degree=6),
                                              random_state=4),
                   n_jobs=-1,
                   param_distributions={'max_features': [0.5, 0.75, 1.0],
                                        'max_samples': [0.5, 0.75, 1.0]},
                   scoring='neg_root_mean_squared_error', verbose=1)
print("The best BaggingRegressor coarse grid parameters:", opt_grid.best_params_)

best_rmse = 10000
best_r2 = 0
for estimator in opt_grid.best_estimator_.estimators_:
    if np.sqrt(estimator.mse_) < best_rmse:
        best_rmse = np.sqrt(estimator.mse_)
    if estimator.rsq_ > best_r2:
        best_r2 = estimator.rsq_
The best BaggingRegressor coarse grid parameters: {'max_samples': 1.0, 'max_features': 1.0}
print("Tuned BaggingRegressor Train RMSE:", best_rmse)
print("Tuned BaggingRegressor R2:", best_r2)
print("Tuned BaggingRegressor Test RMSE: ", mean_squared_error(y_test, opt_grid.predict(X_test), squared = False))
print("Tuned BaggingRegressor R2", r2_score(y_test, opt_grid.predict(X_test)))
Tuned BaggingRegressor Train RMSE: 2263.553508108404
Tuned BaggingRegressor R2: 0.9721508909839748
Tuned BaggingRegressor Test RMSE:  2615.7244744165496
Tuned BaggingRegressor R2 0.8086380025887268

Analysis of Results

The best RMSE and R2 results from the tuned bagging model on the best MARS model. The best RMSE is 2615.7 and the best R2 is 0.81

2.2 Decision Trees & Bagging Decision Trees

By Nicole Birova

2.2.1 Base Decision (Regression) Tree

base = DecisionTreeRegressor(random_state = 4, max_depth = 3)
base.fit(X_train_mars, y_train_mars)

y_pred = base.predict(X_test_mars)

mse = mean_squared_error(y_test_mars, y_pred)
rmse = np.sqrt(mean_squared_error(y_test_mars, y_pred))
print("Mean Squared Error:", mse)
print("RMSE: ", rmse)
Mean Squared Error: 15049713.713885518
RMSE:  3879.3960501456304
#Libraries for visualizing trees
from sklearn.tree import export_graphviz 
from six import StringIO
from IPython.display import Image  
import pydotplus
import time as tm
#Visualizing the regression tree
dot_data = StringIO()
export_graphviz(base, out_file=dot_data,  
                filled=True, rounded=True,
                feature_names =['model_name', 'mileage', 'power', 'year', 'wheel_system_cat', 'engine_cylinders_V6_Flex_Fuel_Vehicle', 'fleet', 'savings_amount'])
graph = pydotplus.graph_from_dot_data(dot_data.getvalue())  
graph.write_png('car_price_tree.png')
Image(graph.create_png())

2.2.2 Coarse Grid Tuning

model = DecisionTreeRegressor(random_state=4) 
model.fit(X, y)
print("Maximum tree depth =", model.get_depth())
print("Maximum leaves =", model.get_n_leaves())
Maximum tree depth = 32
Maximum leaves = 4485
#Finding cross-validation error for trees 
parameters = {'max_depth':range(2,35, 3),'max_leaf_nodes':range(2,5000, 100)}
cv = KFold(n_splits = 5,shuffle=True,random_state=4)
model = GridSearchCV(DecisionTreeRegressor(random_state=4), parameters, n_jobs=-1,verbose=1,cv=cv,
                    scoring=['neg_mean_squared_error', 'r2'], refit = 'neg_mean_squared_error')
model.fit(X, y)
print (model.best_score_, model.best_params_) 
Fitting 5 folds for each of 550 candidates, totalling 2750 fits
-7826521.035438128 {'max_depth': 17, 'max_leaf_nodes': 202}

From the above cross-validation, the optimal hyperparameter values are max_depth = 17 and max_leaf_nodes = 202.

#Detailed results of k-fold cross validation
cv_results = pd.DataFrame(model.cv_results_)
#cv_results.head()
fig, axes = plt.subplots(1,2,figsize=(14,5))
plt.subplots_adjust(wspace=0.2)
axes[0].plot(cv_results.param_max_depth, np.sqrt(-cv_results.mean_test_neg_mean_squared_error), 'o')
axes[0].set_ylim([2750, 3250])
axes[0].set_xlabel('Depth')
axes[0].set_ylabel('K-fold RMSE')
axes[1].plot(cv_results.param_max_leaf_nodes, np.sqrt(-cv_results.mean_test_neg_mean_squared_error), 'o')
axes[1].set_ylim([2750, 3050])
axes[1].set_xlabel('Leaves')
axes[1].set_ylabel('K-fold RMSE');

We observe that for a depth of around 10 to 15, and number of leaves within 1500, we get the lowest K-fold RMSE. So, we do a finer search in that region to obtain more precise hyperparameter values.

#Developing the tree based on optimal hyperparameters found by Course Grid 
model = DecisionTreeRegressor(random_state=4, max_depth=17,max_leaf_nodes=202) 
model.fit(X_train, y_train)
DecisionTreeRegressor(max_depth=17, max_leaf_nodes=202, random_state=4)
#RMSE on test data for Course Grid
np.sqrt(mean_squared_error(y_test, model.predict(X_test)))
2937.5060204053157

2.2.4 Cost-Complexity Pruning

CCPmodel = DecisionTreeRegressor(random_state = 4)#model without any restrictions
path= CCPmodel.cost_complexity_pruning_path(X_train,y_train)# Compute pruning path
alphas=path['ccp_alphas']
len(alphas)
2886
#start_time = tm.time()
cv = KFold(n_splits = 5,shuffle=True,random_state=1)
tree = GridSearchCV(DecisionTreeRegressor(random_state=4), param_grid = {'ccp_alpha':alphas}, 
                     scoring = 'neg_mean_squared_error',n_jobs=-1,verbose=1,cv=cv)
tree.fit(X_train, y_train)
print (tree.best_score_, tree.best_params_)
#print("Time taken =",round((tm.time()-start_time)/60), "minutes")
Fitting 5 folds for each of 2886 candidates, totalling 14430 fits
-7937206.2704411475 {'ccp_alpha': 31132.093956765573}
model = DecisionTreeRegressor(ccp_alpha=16584.831558986974,random_state=4)
model.fit(X_train, y_train)
pred = model.predict(X_test)
np.sqrt(mean_squared_error(y_test, pred))
2924.7599538817894
gridcv_results = pd.DataFrame(tree.cv_results_)
cv_error = -gridcv_results['mean_test_score']
#Visualizing the 5-fold cross validation error vs alpha
plt.plot(alphas,cv_error)
plt.xscale('log')
plt.xlabel('alpha')
plt.ylabel('K-fold MSE');

#Zooming in to see the alpha where the 5-fold cross validation error is minimizing
plt.plot(alphas[0:4000],cv_error[0:4000])
plt.xlabel('alpha')
plt.ylabel('K-fold MSE');

#stime = tm.time()
trees=[]
for i in alphas:
    tree = DecisionTreeRegressor(ccp_alpha=i,random_state=4)
    tree.fit(X_train, y_train)
    trees.append(tree)
#print(tm.time()-stime)
node_counts = [clf.tree_.node_count for clf in trees]
depth = [clf.tree_.max_depth for clf in trees]

fig, ax = plt.subplots(1, 2,figsize=(10,6))
ax[0].plot(alphas[0:5000], node_counts[0:5000], marker="o", drawstyle="steps-post")
ax[0].set_xlabel("alpha")
ax[0].set_ylabel("number of nodes")
ax[0].set_title("Number of nodes vs alpha")
ax[1].plot(alphas[0:5500], depth[0:5500], marker="o", drawstyle="steps-post")
ax[1].set_xlabel("alpha")
ax[1].set_ylabel("depth of tree")
ax[1].set_title("Depth vs alpha")
Text(0.5, 1.0, 'Depth vs alpha')

train_scores = [clf.score(X_train, y_train) for clf in trees]
test_scores = [clf.score(X_test, y_test) for clf in trees]
fig, ax = plt.subplots()
ax.set_xlabel("alpha")
ax.set_ylabel("accuracy")
ax.set_title("Accuracy vs alpha for training and testing sets")
ax.plot(alphas[0:5450], train_scores[0:5450], marker="o", label="train", drawstyle="steps-post")#Plotting the zoomed-in plot (ignoring very high alphas), otherwise it is hard to see the trend
ax.plot(alphas[0:5450], test_scores[0:5450], marker="o", label="test", drawstyle="steps-post")#Plotting the zoomed-in plot (ignoring very high alphas), otherwise it is hard to see the trend
ax.legend()
plt.show()

2.2.5 Bagging

#Bagging the results of 1 decision tree (just to see impact in RMSE)
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=1, random_state=4,n_jobs=-1).fit(X_train, y_train)
np.sqrt(mean_squared_error(y_test, model.predict(X_test)))
3508.2134690063613
#Bagging the results of 10 decision trees
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=10, random_state=4,n_jobs=-1).fit(X_train, y_train)
np.sqrt(mean_squared_error(y_test, model.predict(X_test)))
2427.7266689078633
#Bagging the results of 100 decision trees
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=100, random_state=4,n_jobs=-1).fit(X_train, y_train)
np.sqrt(mean_squared_error(y_test, model.predict(X_test)))
2316.262383026707
#Finding model accuracy vs number of trees
oob_rsquared={};test_rsquared={};oob_rmse={};test_rmse = {}
for i in np.linspace(10,400,40,dtype=int):
    model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=i, random_state=4,
                        n_jobs=-1,oob_score=True).fit(X, y)
    oob_rsquared[i]=model.oob_score_  #Returns the out-of_bag R-squared 
    test_rsquared[i]=model.score(X_test,y_test) #Returns the test R-squared 
    oob_rmse[i]=np.sqrt(mean_squared_error(model.oob_prediction_,y))
    test_rmse[i]=np.sqrt(mean_squared_error(model.predict(X_test),y_test))
plt.rcParams.update({'font.size': 15})
plt.figure(figsize=(8, 6), dpi=80)
plt.plot(oob_rsquared.keys(),oob_rsquared.values(),label = 'Out of bag R-squared')
plt.plot(oob_rsquared.keys(),oob_rsquared.values(),'o',color = 'blue')
plt.plot(test_rsquared.keys(),test_rsquared.values(), label = 'Test data R-squared')
plt.xlabel('Number of trees')
plt.ylabel('Rsquared')
plt.legend()
<matplotlib.legend.Legend at 0x7fb5a96595e0>

#Visualizing out-of-bag RMSE and test data RMSE
plt.rcParams.update({'font.size': 15})
plt.figure(figsize=(8, 6), dpi=80)
plt.plot(oob_rmse.keys(),oob_rmse.values(),label = 'Out of bag RMSE')
plt.plot(oob_rmse.keys(),oob_rmse.values(),'o',color = 'blue')
plt.plot(test_rmse.keys(),test_rmse.values(), label = 'Test data RMSE')
plt.xlabel('Number of trees')
plt.ylabel('RMSE')
plt.legend()
<matplotlib.legend.Legend at 0x7fb5a971cc40>

The out-of-bag R-squared initially increases, and then stabilizes after a certain number of trees (around 50 in this case). Similarly, the RMSE follows the same trend.

#Bagging with 50 trees
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=50, random_state=4,
                        oob_score=True,n_jobs=-1).fit(X_train, y_train)
model.oob_score_
0.862093480612047
#RMSE on test data
pred = model.predict(X_test)
np.sqrt(mean_squared_error(y_test, pred))
2344.7245506909007
#Optimizing bagging hyperparameters using grid search
n_samples = X_train.shape[0]
n_features = 20

params = {'base_estimator': [DecisionTreeRegressor(random_state = 4),LinearRegression()],
          'n_estimators': [50],
          'max_samples': [0.5,1.0],
          'max_features': [0.5,1.0],
          'bootstrap': [True, False],
          'bootstrap_features': [True, False]}

cv = KFold(n_splits=5,shuffle=True,random_state=1)
bagging_regressor_grid = GridSearchCV(BaggingRegressor(random_state=4, n_jobs=-1), 
                                      param_grid =params, cv=cv, n_jobs=-1, verbose=1)
bagging_regressor_grid.fit(X_train, y_train)

print('Train R^2 Score : %.3f'%bagging_regressor_grid.best_estimator_.score(X_train, y_train))
print('Test R^2 Score : %.3f'%bagging_regressor_grid.best_estimator_.score(X_test, y_test))
print('Best R^2 Score Through Grid Search : %.3f'%bagging_regressor_grid.best_score_)
print('Best Parameters : ',bagging_regressor_grid.best_params_)
Fitting 5 folds for each of 32 candidates, totalling 160 fits
Train R^2 Score : 0.999
Test R^2 Score : 0.862
Best R^2 Score Through Grid Search : 0.867
Best Parameters :  {'base_estimator': DecisionTreeRegressor(random_state=4), 'bootstrap': False, 'bootstrap_features': False, 'max_features': 0.5, 'max_samples': 1.0, 'n_estimators': 50}
#Model with optimal hyperparameters 
model = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=50, random_state=4,
                        oob_score=True,n_jobs=-1,bootstrap_features=True,bootstrap=True,
                        max_features=1.0,max_samples=1.0).fit(X_train, y_train)
#RMSE on test data
np.sqrt(mean_squared_error(y_test, model.predict(X_test)))
2238.2224784779655

Bagging showed to be the best way to reduce RMSE, producing a lower RMSE than all the other decision tree models.

2.3 Random forest

By Hiba Khatib

The first set of Random Forests will be performed on the MARS selected features.

base_rf = RandomForestRegressor(random_state = 4).fit(X_train_mars, y_train_mars)
print("Base RandomForest Test RMSE: ", mean_squared_error(y_test_mars, base_rf.predict(X_test_mars), squared = False))
print("Base RandomForest Train RMSE: ", mean_squared_error(y_train_mars, base_rf.predict(X_train_mars), squared = False))
print("Base RandomForest R^2: ", r2_score(y_test_mars, base_rf.predict(X_test_mars)))
Base RandomForest Test RMSE:  2369.927998736765
Base RandomForest Train RMSE:  936.6562391425674
Base RandomForest R^2:  0.840005202649853
base_rf_feats = pd.DataFrame(zip(base_rf.feature_names_in_, base_rf.feature_importances_)).reset_index()
base_rf_feats.sort_values(by = base_rf_feats.columns[2], ascending = False)
index 0 1
3 3 year 0.343984
0 0 model_name 0.242288
2 2 power 0.169672
1 1 mileage 0.147431
7 7 savings_amount 0.055077
4 4 wheel_system_cat 0.030605
6 6 fleet 0.005677
5 5 engine_cylinders_V6_Flex_Fuel_Vehicle 0.005266

2.3.2 Fine Grid Search

rf_params2 = {'n_estimators': [400, 700, 900], 
            'max_depth': [6, 8, 10], 
            'max_leaf_nodes': [250, 300]}

rf_grid2 = GridSearchCV(RandomForestRegressor(random_state = 4, bootstrap = True), rf_params2, n_jobs = -1, verbose = 1, 
                       scoring = 'neg_root_mean_squared_error')
rf_grid2.fit(X_train_mars, y_train_mars)
Fitting 5 folds for each of 18 candidates, totalling 90 fits
GridSearchCV(estimator=RandomForestRegressor(random_state=4), n_jobs=-1,
             param_grid={'max_depth': [6, 8, 10], 'max_leaf_nodes': [250, 300],
                         'n_estimators': [400, 700, 900]},
             scoring='neg_root_mean_squared_error', verbose=1)
print("Best RandomForest parameters:", rf_grid2.best_params_)
print("Best model train RMSE:", mean_squared_error(y_train_mars, rf_grid2.predict(X_train_mars), squared = False))
print("Best model test RMSE:", mean_squared_error(y_test_mars, rf_grid2.predict(X_test_mars), squared = False))
print("Best model R^2:", r2_score(y_test_mars, rf_grid2.predict(X_test_mars)))
rf_feats2 = pd.DataFrame(zip(rf_grid2.best_estimator_.feature_names_in_, rf_grid2.best_estimator_.feature_importances_)).reset_index()
rf_feats2.sort_values(by = rf_feats2.columns[2], ascending = False)
Best RandomForest parameters: {'max_depth': 10, 'max_leaf_nodes': 300, 'n_estimators': 400}
Best model train RMSE: 1760.3116869136927
Best model test RMSE: 2411.105836013617
Best model R^2: 0.8343970360415414
index 0 1
3 3 year 0.366251
0 0 model_name 0.264704
2 2 power 0.163420
1 1 mileage 0.134579
7 7 savings_amount 0.032096
4 4 wheel_system_cat 0.029813
5 5 engine_cylinders_V6_Flex_Fuel_Vehicle 0.005402
6 6 fleet 0.003734

Now we will continue with analysis on our selected features

base_rf = RandomForestRegressor(random_state = 4).fit(X_train, y_train)
print("Base RandomForest Test RMSE: ", mean_squared_error(y_test, base_rf.predict(X_test), squared = False))
print("Base RandomForest Train RMSE: ", mean_squared_error(y_train, base_rf.predict(X_train), squared = False))
print("Base RandomForest R^2: ", r2_score(y_test, base_rf.predict(X_test)))
Base RandomForest Test RMSE:  2320.299016980909
Base RandomForest Train RMSE:  811.3938236091882
Base RandomForest R^2:  0.8494226618329384
base_rf_feats = pd.DataFrame(zip(base_rf.feature_names_in_, base_rf.feature_importances_)).reset_index()
base_rf_feats.sort_values(by = base_rf_feats.columns[2], ascending = False)
index 0 1
13 13 trimId 0.367903
8 8 model_name 0.206468
10 10 power 0.117995
7 7 mileage 0.109063
12 12 torque 0.028295
16 16 year 0.025966
11 11 savings_amount 0.024256
4 4 height 0.015664
6 6 length 0.013655
5 5 highway_fuel_economy 0.011459
3 3 fuel_tank_volume 0.011209
15 15 width 0.011103
21 21 engine_cylinders_V6 0.010807
14 14 wheelbase 0.008779
18 18 wheel_system_display_cat 0.008285
17 17 wheel_system_cat 0.006988
0 0 city_fuel_economy 0.006192
1 1 engine_displacement 0.005792
2 2 franchise_make 0.004500
9 9 owner_count 0.003003
22 22 transmission_display_8_Speed_Automatic 0.001193
19 19 body_type_Sedan 0.000981
20 20 engine_cylinders_I4 0.000442 
#### Coarse GridSearch
rf_params = {'n_estimators': [250, 500, 700], 
            'max_depth': [4, 6, 8], 
            'max_leaf_nodes': range(2, 300, 75),
            'bootstrap': [True, False]}

rf_grid = GridSearchCV(RandomForestRegressor(random_state = 4), rf_params, n_jobs = -1, verbose = 1, 
                       scoring = 'neg_root_mean_squared_error')
rf_grid.fit(X_train, y_train)

print("Best RandomForest parameters:", rf_grid.best_params_)
print("Best model train RMSE:", mean_squared_error(y_train, rf_grid.predict(X_train), squared = False))
print("Best model test RMSE:", mean_squared_error(y_test, rf_grid.predict(X_test), squared = False))
print("Best model R^2:", r2_score(y_test, rf_grid.predict(X_test)))

rf_feats = pd.DataFrame(zip(rf_grid.best_estimator_.feature_names_in_, rf_grid.best_estimator_.feature_importances_)).reset_index()
rf_feats.sort_values(by = rf_feats.columns[2], ascending = False)
Fitting 5 folds for each of 72 candidates, totalling 360 fits
Best RandomForest parameters: {'bootstrap': True, 'max_depth': 8, 'max_leaf_nodes': 227, 'n_estimators': 700}
Best model train RMSE: 1798.8381156882924
Best model test RMSE: 2515.1964258302187
Best model R^2: 0.8230642798700507
index 0 1
13 13 trimId 0.397924
8 8 model_name 0.227093
10 10 power 0.122868
7 7 mileage 0.102456
12 12 torque 0.028858
16 16 year 0.022507
4 4 height 0.011860
21 21 engine_cylinders_V6 0.010883
6 6 length 0.010647
11 11 savings_amount 0.008886
5 5 highway_fuel_economy 0.008685
15 15 width 0.008171
3 3 fuel_tank_volume 0.007906
17 17 wheel_system_cat 0.007112
18 18 wheel_system_display_cat 0.007047
14 14 wheelbase 0.006170
1 1 engine_displacement 0.003463
0 0 city_fuel_economy 0.003062
9 9 owner_count 0.001666
2 2 franchise_make 0.001565
22 22 transmission_display_8_Speed_Automatic 0.000580
19 19 body_type_Sedan 0.000419
20 20 engine_cylinders_I4 0.000169 
fig, axes = plt.subplots(1,3,figsize=(14,5))
plt.subplots_adjust(wspace=0.2)
axes[0].plot(rf_results.param_max_leaf_nodes, (-rf_results.mean_test_score), 'o')
axes[0].set_xlabel('max leaf nodes')
axes[0].set_ylabel('MAE')
axes[1].plot(rf_results.param_n_estimators,(-rf_results.mean_test_score), 'o')
axes[1].set_xlabel('n estimators')
axes[1].set_ylabel('MAE');
axes[2].plot(rf_results.param_max_depth,(-rf_results.mean_test_score), 'o')
axes[2].set_xlabel('max_depth')
axes[2].set_ylabel('MAE');

#### fine grid search
rf_params2 = {'n_estimators': [500, 700, 900], 
            'max_depth': [6, 8, 10], 
            'max_leaf_nodes': range(250, 350, 15)}

rf_grid2 = GridSearchCV(RandomForestRegressor(random_state = 4, bootstrap = True), rf_params2, n_jobs = -1, verbose = 1, 
                       scoring = 'neg_root_mean_squared_error')
rf_grid2.fit(X_train, y_train)

print("Best RandomForest parameters:", rf_grid2.best_params_)
print("Best model train RMSE:", mean_squared_error(y_train, rf_grid2.predict(X_train), squared = False))
print("Best model test RMSE:", mean_squared_error(y_test, rf_grid2.predict(X_test), squared = False))
print("Best model R^2:", r2_score(y_test, rf_grid2.predict(X_test)))
rf_feats2 = pd.DataFrame(zip(rf_grid2.best_estimator_.feature_names_in_, rf_grid2.best_estimator_.feature_importances_)).reset_index()
rf_feats2.sort_values(by = rf_feats2.columns[2], ascending = False)
Fitting 5 folds for each of 63 candidates, totalling 315 fits
Best RandomForest parameters: {'max_depth': 10, 'max_leaf_nodes': 340, 'n_estimators': 500}
Best model train RMSE: 1424.986609472238
Best model test RMSE: 2385.598110495618
Best model R^2: 0.8408281485903418
index 0 1
13 13 trimId 0.384392
8 8 model_name 0.216342
10 10 power 0.119587
7 7 mileage 0.105961
12 12 torque 0.030705
16 16 year 0.022599
4 4 height 0.014340
11 11 savings_amount 0.014279
6 6 length 0.012538
21 21 engine_cylinders_V6 0.010676
15 15 width 0.010466
5 5 highway_fuel_economy 0.009993
3 3 fuel_tank_volume 0.009782
14 14 wheelbase 0.007699
18 18 wheel_system_display_cat 0.007544
17 17 wheel_system_cat 0.007222
1 1 engine_displacement 0.004716
0 0 city_fuel_economy 0.004385
2 2 franchise_make 0.002616
9 9 owner_count 0.002076
22 22 transmission_display_8_Speed_Automatic 0.001105
19 19 body_type_Sedan 0.000651
20 20 engine_cylinders_I4 0.000327

The best RMSE and R^2 resulted from the fine random forest grid search using our feature selection. This is because our feature selection was done using random forest.

Our best random forest RMSE and R2 are 2385.598110495618 and 0.8408281485903418 respectively.

2.4 CatBoost

By YuYan Zhang

import catboost as cb
from catboost import CatBoostRegressor
train_pool = cb.Pool(X_train,y_train)
test_pool = cb.Pool(X_test,y_test)
cb1 = CatBoostRegressor(loss_function='RMSE')
cb1.fit(train_pool)
Learning rate set to 0.049132
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<catboost.core.CatBoostRegressor at 0x14b563f2520>
# train
print("Train RMSE:", mean_squared_error(y_train, cb1.predict(train_pool), squared = False))
print("Train R-squared:",r2_score(y_train, cb1.predict(train_pool)))
Train RMSE: 1199.8271873207266
Train R-squared: 0.9597114208072007
# test
print("Test RMSE:", mean_squared_error(y_test, cb1.predict(test_pool), squared = False))
print("Test R-squared:",r2_score(y_test, cb1.predict(test_pool)))
Test RMSE: 2055.989006548227
Test R-squared: 0.8817739227682351
feature_importance = cb1.feature_importances_
sorted_idx = np.argsort(feature_importance)
fig = plt.figure(figsize=(12, 6))
plt.barh(range(len(sorted_idx)), feature_importance[sorted_idx], align='center')
plt.yticks(range(len(sorted_idx)), np.array(X_test.columns)[sorted_idx])
plt.title('Feature Importance')
Text(0.5, 1.0, 'Feature Importance')

cb2 = CatBoostRegressor()

cb_params = {'learning_rate':[0.01, 0.1],
         'depth': [4, 7, 10],
         'l2_leaf_reg': [2, 6, 10],
         'random_strength': [0, 5, 10]
}

cb_search_result = cb2.grid_search(cb_params, 
                                       X=X_train, 
                                       y=y_train, 
                                       plot=True)
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bestTest = 2029.16279
bestIteration = 724

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bestIteration = 999

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bestIteration = 988

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bestTest = 2366.205888
bestIteration = 999

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bestTest = 2050.022234
bestIteration = 993

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bestTest = 2073.951938
bestIteration = 543

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bestTest = 2504.001754
bestIteration = 999

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bestIteration = 927

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bestTest = 2533.408781
bestIteration = 999

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bestTest = 2050.266293
bestIteration = 997

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bestTest = 2043.371214
bestIteration = 716

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bestTest = 2590.898534
bestIteration = 999

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bestTest = 2050.301596
bestIteration = 982

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bestTest = 2606.463862
bestIteration = 999

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bestTest = 2049.768502
bestIteration = 825

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bestIteration = 402

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bestTest = 2180.146244
bestIteration = 999

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bestTest = 2026.048825
bestIteration = 447

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bestTest = 2243.903425
bestIteration = 999

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bestTest = 2042.682461
bestIteration = 607

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bestTest = 2337.617359
bestIteration = 999

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bestTest = 2015.321625
bestIteration = 543

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bestTest = 2393.572228
bestIteration = 999

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bestIteration = 646

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bestTest = 1997.750804
bestIteration = 504

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bestTest = 2417.706563
bestIteration = 999

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bestTest = 2005.795186
bestIteration = 827

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bestIteration = 923

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bestTest = 2125.481267
bestIteration = 999

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bestTest = 2088.55228
bestIteration = 175

37: loss: 2088.5522803  best: 1997.7508039 (31) total: 2m 45s   remaining: 1m 9s
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bestTest = 2137.424331
bestIteration = 999

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bestTest = 2085.537619
bestIteration = 336

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bestTest = 2180.205334
bestIteration = 999

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bestTest = 2062.723088
bestIteration = 342

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bestTest = 2260.234487
bestIteration = 999

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bestTest = 2148.005408
bestIteration = 532

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bestIteration = 548

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bestIteration = 516

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bestTest = 2148.892497
bestIteration = 436

49: loss: 2148.8924969  best: 1997.7508039 (31) total: 12m 26s  remaining: 59.8s
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1:  learn: 19727.5691607    test: 19955.2503686 best: 19955.2503686 (1) total: 72.1ms   remaining: 36s
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bestTest = 2413.880145
bestIteration = 999

50: loss: 2413.8801453  best: 1997.7508039 (31) total: 13m 11s  remaining: 46.6s
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bestTest = 2126.365609
bestIteration = 715

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bestTest = 2484.54212
bestIteration = 999

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bestTest = 2142.579273
bestIteration = 696

53: loss: 2142.5792731  best: 1997.7508039 (31) total: 15m 37s  remaining: 0us
Estimating final quality...
Training on fold [0/3]
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bestTest = 2110.494946
bestIteration = 342

Training on fold [1/3]
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bestTest = 2163.707912
bestIteration = 433

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bestTest = 2105.196343
bestIteration = 295

After visualizing the GridSearch results, we found that there continues to be the trend of way higher performance on train data, while test performance hovered around the same range. Therefore, we decided keep the untuned model.

2.5 XGBoost

By Emily Leibfritz

import time
from xgboost import XGBRegressor
start_time = time.time()
param_grid = {'max_depth': [8],
              'learning_rate': [0.08,0.09,0.1],
               'reg_lambda':[0.09],
                'n_estimators':[510,515,520,530],
                'gamma': [0.1,0.2],
                'subsample': [0.75,0.8],
                'colsample_bytree': [0.75,0.8]}

cv = KFold(n_splits=5,shuffle=True,random_state=4)
optimal_params = GridSearchCV(estimator=XGBRegressor(random_state=4),                                                       
                             param_grid = param_grid,
                             verbose = 1,
                             n_jobs=-1,
                             cv = cv)
optimal_params.fit(X_train,y_train)
print("Optimal parameter values =", optimal_params.best_params_)
print("Optimal cross validation R-squared = ",optimal_params.best_score_)
print("Time taken = ", round((time.time()-start_time)/60), " minutes")
Fitting 5 folds for each of 96 candidates, totalling 480 fits
Optimal parameter values = {'colsample_bytree': 0.75, 'gamma': 0.2, 'learning_rate': 0.08, 'max_depth': 8, 'n_estimators': 510, 'reg_lambda': 0.09, 'subsample': 0.8}
Optimal cross validation R-squared =  0.8740821137070345
Time taken =  11  minutes
np.sqrt(mean_squared_error(optimal_params.best_estimator_.predict(X_test),y_test))
2171.06738933653
XGBoost_model = XGBRegressor(random_state = 4, colsample_bytree =0.8, gamma = 0.2, learning_rate = 0.09, 
                                 max_depth = 8, n_estimators = 510, reg_lambda = 0.09, subsample = 0.8)
XGBoost_model.fit(X_train,y_train,eval_set = ([(X_test, y_test)]), early_stopping_rounds = 10)
[0] validation_0-rmse:18343.24339
[1] validation_0-rmse:16730.82889
[2] validation_0-rmse:15276.12695
[3] validation_0-rmse:13960.30444
[4] validation_0-rmse:12756.05042
[5] validation_0-rmse:11671.49624
[6] validation_0-rmse:10691.13908
[7] validation_0-rmse:9793.15786
[8] validation_0-rmse:8972.68909
[9] validation_0-rmse:8239.99426
[10]    validation_0-rmse:7570.86028
[11]    validation_0-rmse:6955.64922
[12]    validation_0-rmse:6410.91039
[13]    validation_0-rmse:5921.50069
[14]    validation_0-rmse:5486.91019
[15]    validation_0-rmse:5093.51123
[16]    validation_0-rmse:4743.50895
[17]    validation_0-rmse:4432.95049
[18]    validation_0-rmse:4150.02788
[19]    validation_0-rmse:3901.96935
[20]    validation_0-rmse:3685.42492
[21]    validation_0-rmse:3487.92996
[22]    validation_0-rmse:3313.88922
[23]    validation_0-rmse:3163.10376
[24]    validation_0-rmse:3030.47686
[25]    validation_0-rmse:2912.89194
[26]    validation_0-rmse:2806.51288
[27]    validation_0-rmse:2720.81462
[28]    validation_0-rmse:2646.72958
[29]    validation_0-rmse:2581.43259
[30]    validation_0-rmse:2526.71859
[31]    validation_0-rmse:2478.77626
[32]    validation_0-rmse:2440.19348
[33]    validation_0-rmse:2400.50806
[34]    validation_0-rmse:2367.45884
[35]    validation_0-rmse:2340.97622
[36]    validation_0-rmse:2317.81497
[37]    validation_0-rmse:2297.84136
[38]    validation_0-rmse:2279.19623
[39]    validation_0-rmse:2258.57359
[40]    validation_0-rmse:2247.05785
[41]    validation_0-rmse:2236.07640
[42]    validation_0-rmse:2230.09781
[43]    validation_0-rmse:2222.03256
[44]    validation_0-rmse:2214.23880
[45]    validation_0-rmse:2210.15261
[46]    validation_0-rmse:2206.67277
[47]    validation_0-rmse:2201.42420
[48]    validation_0-rmse:2198.51972
[49]    validation_0-rmse:2193.38451
[50]    validation_0-rmse:2189.72437
[51]    validation_0-rmse:2186.85459
[52]    validation_0-rmse:2184.37442
[53]    validation_0-rmse:2181.43863
[54]    validation_0-rmse:2180.96694
[55]    validation_0-rmse:2178.10353
[56]    validation_0-rmse:2178.61863
[57]    validation_0-rmse:2178.67483
[58]    validation_0-rmse:2176.07221
[59]    validation_0-rmse:2175.93886
[60]    validation_0-rmse:2175.40200
[61]    validation_0-rmse:2173.80754
[62]    validation_0-rmse:2171.90952
[63]    validation_0-rmse:2168.83789
[64]    validation_0-rmse:2169.38154
[65]    validation_0-rmse:2167.27993
[66]    validation_0-rmse:2168.37356
[67]    validation_0-rmse:2167.99127
[68]    validation_0-rmse:2167.68498
[69]    validation_0-rmse:2167.30236
[70]    validation_0-rmse:2166.63438
[71]    validation_0-rmse:2166.43820
[72]    validation_0-rmse:2166.83759
[73]    validation_0-rmse:2166.03560
[74]    validation_0-rmse:2161.70525
[75]    validation_0-rmse:2161.03648
[76]    validation_0-rmse:2160.75528
[77]    validation_0-rmse:2159.48954
[78]    validation_0-rmse:2160.03194
[79]    validation_0-rmse:2160.20564
[80]    validation_0-rmse:2161.14530
[81]    validation_0-rmse:2162.02055
[82]    validation_0-rmse:2159.29397
[83]    validation_0-rmse:2160.20823
[84]    validation_0-rmse:2159.92115
[85]    validation_0-rmse:2159.58211
[86]    validation_0-rmse:2158.79554
[87]    validation_0-rmse:2158.64334
[88]    validation_0-rmse:2158.08751
[89]    validation_0-rmse:2158.72923
[90]    validation_0-rmse:2159.28733
[91]    validation_0-rmse:2160.65406
[92]    validation_0-rmse:2160.29895
[93]    validation_0-rmse:2160.70149
[94]    validation_0-rmse:2159.11506
[95]    validation_0-rmse:2158.94153
[96]    validation_0-rmse:2158.65374
[97]    validation_0-rmse:2158.87579
[98]    validation_0-rmse:2159.12650
XGBRegressor(base_score=None, booster=None, callbacks=None,
             colsample_bylevel=None, colsample_bynode=None,
             colsample_bytree=0.8, early_stopping_rounds=None,
             enable_categorical=False, eval_metric=None, feature_types=None,
             gamma=0.2, gpu_id=None, grow_policy=None, importance_type=None,
             interaction_constraints=None, learning_rate=0.09, max_bin=None,
             max_cat_threshold=None, max_cat_to_onehot=None,
             max_delta_step=None, max_depth=8, max_leaves=None,
             min_child_weight=None, missing=nan, monotone_constraints=None,
             n_estimators=510, n_jobs=None, num_parallel_tree=None,
             predictor=None, random_state=4, ...)
XGB_bagged = BaggingRegressor(base_estimator=XGBoost_model, 
                             n_estimators=20,
                             bootstrap=False,
                            random_state=4,
                            n_jobs = -1).fit(X_train, y_train)
np.sqrt(mean_squared_error(XGB_bagged.predict(X_test),y_test))
2088.5007849998374

2.6 LGBM

By Emily Leibfritz

from lightgbm import LGBMRegressor
LGMB_model=LGBMRegressor(random_state=4).fit(X_train,y_train)
np.sqrt(mean_squared_error(LGMB_model.predict(X_test),y_test))
2135.7915997846417
#K-fold cross validation to find optimal parameters for LightGBM regressor
start_time = time.time()
param_grid = {'max_depth': [4,5,6,7,8],
              'num_leaves': [50,60,70,80,90,100,110,120,130,170,200],
              'learning_rate': [0.05, 0.07,0.09,0.1,0.12,0.15],
               'reg_lambda':[4,5,6,7,8,9,10],
                'n_estimators':[800,900,1000,1100,1200,1300,1400,1500],
                'reg_alpha': [1,2,3,4,5,6,7,8,9,10],
                'subsample': [0.70, 0.75,0.8],
                'colsample_bytree': [0.70,0.75,0.8, 0.9]}

cv = KFold(n_splits=5,shuffle=True,random_state=1)
optimal_params = RandomizedSearchCV(estimator=LGBMRegressor(random_state=1),                                                       
                             param_distributions = param_grid, n_iter = 200,
                             verbose = 1,
                             n_jobs=-1,
                             cv = cv)
optimal_params.fit(X_train,y_train)
print("Optimal parameter values =", optimal_params.best_params_)
print("Optimal cross validation R-squared = ",optimal_params.best_score_)
print("Time taken = ", round((time.time()-start_time)/60), " minutes")
Fitting 5 folds for each of 200 candidates, totalling 1000 fits
Optimal parameter values = {'subsample': 0.75, 'reg_lambda': 10, 'reg_alpha': 10, 'num_leaves': 130, 'n_estimators': 1000, 'max_depth': 4, 'learning_rate': 0.05, 'colsample_bytree': 0.7}
Optimal cross validation R-squared =  0.8821169822021782
Time taken =  7  minutes
np.sqrt(mean_squared_error(optimal_params.best_estimator_.predict(X_test),y_test))
2085.4135507703627
LGBM_bagged = BaggingRegressor(base_estimator=optimal_params.best_estimator_, 
                             n_estimators=20,
                             bootstrap=False,
                            random_state=4,
                            n_jobs = -1).fit(X_train, y_train)
np.sqrt(mean_squared_error(LGBM_bagged.predict(X_test),y_test))
2078.6305836054316

3 Model Ensemble

Put code with comments. The comments should explain the code such that it can be easily understood. You may put text (in a markdown cell) before a large chunk of code to explain the overall purpose of the code, if it is not intuitive. Put the name of the person / persons who contributed to each code chunk / set of code chunks.

# models choosen for ensemble
bagged_tree = BaggingRegressor(base_estimator=DecisionTreeRegressor(), n_estimators=50, random_state=4,
                        oob_score=True,n_jobs=-1,bootstrap_features=True,bootstrap=True,
                        max_features=1.0,max_samples=1.0)

LGBM_bagged = BaggingRegressor(base_estimator=LGBMRegressor(subsample= 0.75, reg_lambda=10, 
                                                            reg_alpha= 10, num_leaves=130, 
                                                            n_estimators=1000, max_depth = 4, 
                                                            learning_rate=0.05, colsample_bytree= 0.7), 
                             n_estimators=20,
                             bootstrap=False,
                            random_state=4,
                            n_jobs = -1)

XGB_bagged = BaggingRegressor(base_estimator=XGBoost_model, 
                             n_estimators=20,
                             bootstrap=False,
                            random_state=4,
                            n_jobs = -1)

catboost = CatBoostRegressor(loss_function='RMSE')

3.1 Voting ensemble

en=VotingRegressor(estimators = [('bt',bagged_tree),('lgbm',LGBM_bagged),('xgb',XGB_bagged),('cb',catboost)])
en.fit(X_train,y_train)
Learning rate set to 0.049132
0:  learn: 5794.2995392 total: 60ms remaining: 60s
1:  learn: 5634.2358373 total: 61.3ms   remaining: 30.6s
2:  learn: 5473.4473085 total: 62.9ms   remaining: 20.9s
3:  learn: 5312.7684693 total: 64.3ms   remaining: 16s
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NameError: name 'Xtest' is not defined
print("Ensemble model RMSE = ", np.sqrt(mean_squared_error(en.predict(X_test),y_test)))
Ensemble model RMSE =  2052.099261567051

3.2 Stacking ensemble(s)

stack_1 = StackingRegressor(estimators = [('bt',bagged_tree),('lgbm',LGBM_bagged),('xgb',XGB_bagged),('cb',catboost)],
                     final_estimator=LinearRegression(),                                          
                    cv = KFold(n_splits = 5, shuffle = True, random_state=1))
stack_1.fit(X_train,y_train)
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StackingRegressor(cv=KFold(n_splits=5, random_state=1, shuffle=True),
                  estimators=[('bt',
                               BaggingRegressor(base_estimator=DecisionTreeRegressor(),
                                                bootstrap_features=True,
                                                n_estimators=50, n_jobs=-1,
                                                oob_score=True,
                                                random_state=4)),
                              ('lgbm',
                               BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
                                                                             learning_rate=0.05,
                                                                             max_depth=4,
                                                                             n_estimators=1000,
                                                                             num...
                                                                            max_delta_step=None,
                                                                            max_depth=8,
                                                                            max_leaves=None,
                                                                            min_child_weight=None,
                                                                            missing=nan,
                                                                            monotone_constraints=None,
                                                                            n_estimators=510,
                                                                            n_jobs=None,
                                                                            num_parallel_tree=None,
                                                                            predictor=None,
                                                                            random_state=4, ...),
                                                bootstrap=False,
                                                n_estimators=20, n_jobs=-1,
                                                random_state=4)),
                              ('cb',
                               <catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
                  final_estimator=LinearRegression())
print("Linear regression metamodel RMSE = ", np.sqrt(mean_squared_error(stack_1.predict(X_test),y_test)))
Linear regression metamodel RMSE =  2033.444711920972
stack_2 = StackingRegressor(estimators = [('bt',bagged_tree),('lgbm',LGBM_bagged),('xgb',XGB_bagged),('cb',catboost)],
                     final_estimator=RandomForestRegressor(),                                          
                    cv = KFold(n_splits = 5, shuffle = True, random_state=1))
stack_2.fit(X_train,y_train)
Learning rate set to 0.049132
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StackingRegressor(cv=KFold(n_splits=5, random_state=1, shuffle=True),
                  estimators=[('bt',
                               BaggingRegressor(base_estimator=DecisionTreeRegressor(),
                                                bootstrap_features=True,
                                                n_estimators=50, n_jobs=-1,
                                                oob_score=True,
                                                random_state=4)),
                              ('lgbm',
                               BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
                                                                             learning_rate=0.05,
                                                                             max_depth=4,
                                                                             n_estimators=1000,
                                                                             num...
                                                                            max_delta_step=None,
                                                                            max_depth=8,
                                                                            max_leaves=None,
                                                                            min_child_weight=None,
                                                                            missing=nan,
                                                                            monotone_constraints=None,
                                                                            n_estimators=510,
                                                                            n_jobs=None,
                                                                            num_parallel_tree=None,
                                                                            predictor=None,
                                                                            random_state=4, ...),
                                                bootstrap=False,
                                                n_estimators=20, n_jobs=-1,
                                                random_state=4)),
                              ('cb',
                               <catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
                  final_estimator=RandomForestRegressor())
print("RF metamodel RMSE = ", np.sqrt(mean_squared_error(stack_2.predict(X_test),y_test)))
RF metamodel RMSE =  2127.8992549473046
from sklearn.linear_model import LassoCV
stack_3 = StackingRegressor(estimators = [('bt',bagged_tree),('lgbm',LGBM_bagged),('xgb',XGB_bagged),('cb',catboost)],
                     final_estimator=LassoCV(),                                          
                    cv = KFold(n_splits = 5, shuffle = True, random_state=1))
stack_3.fit(X_train,y_train)
Learning rate set to 0.049132
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StackingRegressor(cv=KFold(n_splits=5, random_state=1, shuffle=True),
                  estimators=[('bt',
                               BaggingRegressor(base_estimator=DecisionTreeRegressor(),
                                                bootstrap_features=True,
                                                n_estimators=50, n_jobs=-1,
                                                oob_score=True,
                                                random_state=4)),
                              ('lgbm',
                               BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
                                                                             learning_rate=0.05,
                                                                             max_depth=4,
                                                                             n_estimators=1000,
                                                                             num...
                                                                            max_cat_to_onehot=None,
                                                                            max_delta_step=None,
                                                                            max_depth=8,
                                                                            max_leaves=None,
                                                                            min_child_weight=None,
                                                                            missing=nan,
                                                                            monotone_constraints=None,
                                                                            n_estimators=510,
                                                                            n_jobs=None,
                                                                            num_parallel_tree=None,
                                                                            predictor=None,
                                                                            random_state=4, ...),
                                                bootstrap=False,
                                                n_estimators=20, n_jobs=-1,
                                                random_state=4)),
                              ('cb',
                               <catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
                  final_estimator=LassoCV())
print("Lasso metamodel RMSE = ", np.sqrt(mean_squared_error(stack_3.predict(X_test),y_test)))
Lasso metamodel RMSE =  2033.435450298843

3.3 Ensemble of ensembled models

voting_en=VotingRegressor(estimators = [('voting',en),('linear_stacking',stack_1)])
voting_en.fit(X_train,y_train)
Learning rate set to 0.049132
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VotingRegressor(estimators=[('voting',
                             VotingRegressor(estimators=[('bt',
                                                          BaggingRegressor(base_estimator=DecisionTreeRegressor(),
                                                                           bootstrap_features=True,
                                                                           n_estimators=50,
                                                                           n_jobs=-1,
                                                                           oob_score=True,
                                                                           random_state=4)),
                                                         ('lgbm',
                                                          BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
                                                                                                        learning_rate=0.05,
                                                                                                        max_depth=4,
                                                                                                        n_estimators=1000,
                                                                                                        num_leaves=130,
                                                                                                        r...
                                                                                                         max_delta_step=None,
                                                                                                         max_depth=8,
                                                                                                         max_leaves=None,
                                                                                                         min_child_weight=None,
                                                                                                         missing=nan,
                                                                                                         monotone_constraints=None,
                                                                                                         n_estimators=510,
                                                                                                         n_jobs=None,
                                                                                                         num_parallel_tree=None,
                                                                                                         predictor=None,
                                                                                                         random_state=4, ...),
                                                                             bootstrap=False,
                                                                             n_estimators=20,
                                                                             n_jobs=-1,
                                                                             random_state=4)),
                                                           ('cb',
                                                            <catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
                                               final_estimator=RandomForestRegressor()))])
print("Voting Ensemble RMSE = ", np.sqrt(mean_squared_error(voting_en.predict(X_test),y_test)))
Voting Ensemble RMSE =  2059.6950529918718
stacking_en=StackingRegressor(estimators = [('voting',en),('linear_stacking',stack_1)],
                             final_estimator=LinearRegression())
stacking_en.fit(X_train,y_train)
Learning rate set to 0.049132
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StackingRegressor(estimators=[('voting',
                               VotingRegressor(estimators=[('bt',
                                                            BaggingRegressor(base_estimator=DecisionTreeRegressor(),
                                                                             bootstrap_features=True,
                                                                             n_estimators=50,
                                                                             n_jobs=-1,
                                                                             oob_score=True,
                                                                             random_state=4)),
                                                           ('lgbm',
                                                            BaggingRegressor(base_estimator=LGBMRegressor(colsample_bytree=0.7,
                                                                                                          learning_rate=0.05,
                                                                                                          max_depth=4,
                                                                                                          n_estimators=1000,
                                                                                                          num_leaves=130...
                                                                                                           max_leaves=None,
                                                                                                           min_child_weight=None,
                                                                                                           missing=nan,
                                                                                                           monotone_constraints=None,
                                                                                                           n_estimators=510,
                                                                                                           n_jobs=None,
                                                                                                           num_parallel_tree=None,
                                                                                                           predictor=None,
                                                                                                           random_state=4, ...),
                                                                               bootstrap=False,
                                                                               n_estimators=20,
                                                                               n_jobs=-1,
                                                                               random_state=4)),
                                                             ('cb',
                                                              <catboost.core.CatBoostRegressor object at 0x7f8f816b1eb0>)],
                                                 final_estimator=RandomForestRegressor()))],
                  final_estimator=LinearRegression())
print("Stacking Ensemble RMSE = ", np.sqrt(mean_squared_error(stacking_en.predict(X_test),y_test)))
Stacking Ensemble RMSE =  2040.0604592567781

Stacking with a Lasso and linear metamodel lead to the best results. Ensembling the ensembles did not lead to better performance.